Finite Group Databasebeta

← #74795

Subgroups — #74795

Order 384.

592 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#924C212yesmaximal, complemented
2C2×#878C212yesmaximal, complemented
3C3×#386C211yesmaximal, characteristic
4#5221C211yesmaximal, characteristic
5#5220C211yesmaximal, characteristic
6#4948C212yesmaximal, complemented
7#4948C212yesmaximal, complemented
8#4656C212yesmaximal, complemented
9#4647C212yesmaximal, complemented
10#2683—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C22×Dic6C2212yes—
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24#924—28no—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#878—28no—
31C3×#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C2×C22⋊Q8—32no—
35C2×C22⋊Q8—32no—
36#396—32no—
37#396—32no—
38#395—32no—
39#390—31no—
40#389—32no—
41#386—31no—
42#386D311yescharacteristic
43C24×C3C2311yescharacteristic
44C24×C3C2311yescharacteristic
45C2×C3×D4—41no—
46C2×C3×D4—41no—
47C2×C3×D4Q814yescomplemented
48C22×C3×C4—24no—
49C22×C3×C4—41no—
50C22×C3×C4C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×Dic3—24no—
54C22×Dic3—41no—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C2×Dic6—24no—
60C2×Dic6—24no—
61C2×Dic6—42no—
62C3×C22⋊C4—24no—
63C3×C22⋊C4—41no—
64#217—24no—
65#217—24no—
66#217—24no—
67#217—41no—
68C4⋊Dic3—24no—
69C4⋊Dic3—24no—
70C4⋊Dic3—41no—
71Dic3⋊C4—24no—
72Dic3⋊C4—24no—
73Dic3⋊C4—42no—
74Dic3⋊C4D418yescomplemented
75C4×Dic3—41no—
76C22×Q8—32no—
77C22×D4C2×D311yescharacteristic
78C23×C4—31no—
79C23×C4C2×D311yescharacteristic
80C22⋊Q8—68no—
81C22⋊Q8—68no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—31no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4C2×D312yes—
94C2×C22⋊C4C2×D311yescharacteristic
95C2×C42—31no—
96#95—32no—
97#95—32no—
98#95C2×D312yes—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3—21no—
102C23×C3—21no—
103C23×C3—21no—
104C23×C3—21no—
105C23×C3—22no—
106C23×C3—22no—
107C23×C3C2×Q812yes—
108C23×C3C2×Q812yes—
109C23×C3C2411yescharacteristic
110C3×D4—44no—
111C2×C3×C4—22no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—44no—
119C2×C3×C4C2×D414yes—
120C2×C3×C4C2×Q812yes—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—44no—
132C2×Dic3C2×D414yes—
133C2×Dic3C2×D414yes—
134Dic6—48no—
135C24C22×D311yescharacteristic
136C24C22×D311yescharacteristic
137C2×Q8—64no—
138C2×Q8—64no—
139C2×Q8—122no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4Dic614yescomplemented
143C22×C4—24no—
144C22×C4—32no—
145C22×C4—32no—
146C22×C4—32no—
147C22×C4—32no—
148C22×C4—41no—
149C22×C4—64no—
150C22×C4—121no—
151C22×C4C22×D312yes—
152C22×C4C22×D311yescharacteristic
153C22×C4C22×D312yes—
154C4⋊C4—38no—
155C4⋊C4—64no—
156C4⋊C4—64no—
157C4⋊C4—64no—
158C4⋊C4—64no—
159C4⋊C4—122no—
160C4⋊C4—121no—
161C22⋊C4—24no—
162C22⋊C4—41no—
163C22⋊C4—64no—
164C22⋊C4—64no—
165C22⋊C4—64no—
166C22⋊C4—121no—
167C42—121no—
168C22×C3—22no—
169C22×C3—22no—
170C22×C3—24no—
171C22×C3—24no—
172C22×C3—42no—
173C22×C3—42no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—41no—
177C22×C3—41no—
178C22×C3C22×D412yes—
179C22×C3C22×Q811yescharacteristic
180C22×C3H(5,𝔽2)11yescharacteristic
181C22×C3H(5,𝔽2)11yescharacteristic
182C22×C32−1+412yes—
183C3×C4—22no—
184C3×C4—24no—
185C3×C4—42no—
186Dic3—24no—
187Dic3—24no—
188Dic3—42no—
189Dic3—42no—
190C23—22no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23C2×Dic612yes—
199C23C2×Dic612yes—
200C23C23×D311yescharacteristic
201Q8—128no—
202D4—44no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—34no—
211C2×C4—34no—
212C2×C4—44no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—124no—
224C2×C4C2×Dic612yes—
225C2×C4D3×D414yes—
226C2×C3—42no—
227C2×C3—42no—
228C2×C3#53712yes—
229C2×C3#54311yescharacteristic
230C2×C3D4211yescharacteristic
231C2×C3D4×Q812yes—
232C2×C3#56311yescharacteristic
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—42no—
242C22—42no—
243C22C22×Dic611yescharacteristic
244C22C2×D3×D412yes—
245C22#100411yescharacteristic
246C22#100411yescharacteristic
247C22#101012yes—
248C4—22no—
249C4—24no—
250C4—42no—
251C4—64no—
252C4—64no—
253C4—122no—
254C4—122no—
255C3#268311yescharacteristic, complemented, Sylow
256C2—42no—
257C2—42no—
258C2#548311yescharacteristic
259C2D4×Dic612yes—
260C2#559011yescharacteristic
261C2#560812yes—
262C2#565111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC22×C3×D4C22
hypercenter#386D3
hypocenterC3#2683
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#924C2semidirect
2. C2×#878C2semidirect
6. #4948C2semidirect
7. #4948C2semidirect
8. #4656C2semidirect
9. #4647C2semidirect
47. C2×C3×D4Q8semidirect
74. Dic3⋊C4D4semidirect
142. C2×D4Dic6semidirect
255. C3#2683semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#9242
C2×#8782
C3×#3862
#52212
#52202
#49482
#46562
#46472
#26833